In optimization the duality gap between the primal and the dual problems is a measure of the suboptimality of any primal-dual point. In classical mechanics the equations of motion of a system can be derived from the Hamiltonian function, which is a quantity that describes the total energy of the system. In this paper we consider a convex optimization problem consisting of the sum of two convex functions, sometimes referred to as a composite objective, and we identify the duality gap to be the `energy' of the system. In the Hamiltonian formalism the energy is conserved, so we add a contractive term to the standard equations of motion so that this energy decreases linearly (ie, geometrically) with time. This yields a continuous-time ordinary differential equation (ODE) in the primal and dual variables which converges to zero duality gap, ie, optimality.

Multi-UAV cooperative path planning (MUCPP) is a fundamental problem in multi-agent systems, aiming to generate collision-free trajectories for a team of unmanned aerial vehicles (UAVs) to complete distributed tasks efficiently. A key challenge lies in achieving both efficiency, by minimizing total mission cost, and fairness, by balancing the workload among UAVs to avoid overburdening individual agents. This paper presents a novel Iterative Exchange Framework for MUCPP, balancing efficiency and fairness through iterative task exchanges and path refinements. The proposed framework formulates a composite objective that combines the total mission distance and the makespan, and iteratively improves the solution via local exchanges under feasibility and safety constraints. For each UAV, collision-free trajectories are generated using A* search over a terrain-aware configuration space. Comprehensive experiments on multiple terrain datasets demonstrate that the proposed method consistently achieves superior trade-offs between total distance and makespan compared to existing baselines.

This paper was a borderline case that led to significant discussion among the reviewers, but in the end we decided the novel ODE-style approach would be of strong interest to the NeurIPS community. In your revision, please address all promises in the rebuttal and comments in the reviews. If there are additional experiments or examples you can include to demonstrate practical value or application of this technique to machine learning tasks, please include these in the revision to better demonstrate where your method might be used---this was the main weakness identified by the reviewers.

Scientific optimization problems are usually concerned with balancing multiple competing objectives, which come as preferences over both the outcomes of an experiment (e.g. maximize the reaction yield) and the corresponding input parameters (e.g. minimize the use of an expensive reagent). Typically, practical and economic considerations define a hierarchy over these objectives, which must be reflected in algorithms for sample-efficient experiment planning. Herein, we introduce BoTier, a composite objective that can flexibly represent a hierarchy of preferences over both experiment outcomes and input parameters. We provide systematic benchmarks on synthetic and real-life surfaces, demonstrating the robust applicability of BoTier across a number of use cases. Importantly, BoTier is implemented in an auto-differentiable fashion, enabling seamless integration with the BoTorch library, thereby facilitating adoption by the scientific community.

In optimization the duality gap between the primal and the dual problems is a measure of the suboptimality of any primal-dual point. In classical mechanics the equations of motion of a system can be derived from the Hamiltonian function, which is a quantity that describes the total energy of the system. In this paper we consider a convex optimization problem consisting of the sum of two convex functions, sometimes referred to as a composite objective, and we identify the duality gap to be the energy' of the system. In the Hamiltonian formalism the energy is conserved, so we add a contractive term to the standard equations of motion so that this energy decreases linearly (ie, geometrically) with time. This yields a continuous-time ordinary differential equation (ODE) in the primal and dual variables which converges to zero duality gap, ie, optimality.

In optimization the duality gap between the primal and the dual problems is a measure of the suboptimality of any primal-dual point. In classical mechanics the equations of motion of a system can be derived from the Hamiltonian function, which is a quantity that describes the total energy of the system. In this paper we consider a convex optimization problem consisting of the sum of two convex functions, sometimes referred to as a composite objective, and we identify the duality gap to be the energy' of the system. In the Hamiltonian formalism the energy is conserved, so we add a contractive term to the standard equations of motion so that this energy decreases linearly (ie, geometrically) with time. This yields a continuous-time ordinary differential equation (ODE) in the primal and dual variables which converges to zero duality gap, ie, optimality.